2-1
Time Value of Money
Goals
Calculate the value today of cash flows
expected in the future.
Calculate the amount of money needed
to today to generate some future value
of money.
2-2
Time Value of Money
Present values versus future values
Interest rate conversions
Annuities
Perpetuities
Growing cash flows
Amortization (loan) payments
Pricing bonds and stock
2-3
Future Values
0
C0
1
FV1 = C0(1+r)
2
FV2 = FV1(1+r)
= C0(1 + r)2
FVt = C0 × (1 + r)t
How much would $70,000 be worth in 14 years @ 7½%?
FV14 = 70,000(1.0725)14 = $186,492
2-4
Present Values
0
PV = PV1/(1 + r)
= C2/(1 + r)2
1
PV1 = C2/(1 + r)
PV = Ct / (1 + r)t
2
C2
Note that this is the
same formula as for FV
What is the maximum price you would pay today for a machine
that generates a single cash flow of $2,000,000 in 20 years?
Interest rate is 8%
What if you sell
this machine?
PV = 2,000,000/(1.08)20 = $429,096
2-5
Multiple Cash Flows
0
C1
C2
CT
1
2
T
Ct in year t, cash flows last for T years
PV = [C1/(1+r)] + [C2/(1+r)2] +……+ [CT/(1+r)T]
T
PV =  [Ct / (1+r)t]
t=1
2-6
Multiple and Infinite Identical Cash Flows
0
C
C
C
C
1
2
T

Annuity: Finite stream of identical cash
flows
Perpetuity: Infinite stream of identical
cash flows
Identical: separated by an identical
growth rate (g=0 in this example)
2-7
Perpetuities and Annuities
Perpetuity:
C
PV 
r
C
Annuity: PV of annuity 
r

1
1 
t
 (1  r )
How about
future values?
FV of annuity 


C
(1  r ) t  1
r
Understand formula
using timelines



2-8
Growing Cash Flows
Growing Perpetuity
0
C
C(1+g)
C(1+g)2
C(1+g)3
1
2
3
4
Growing Annuity
 Growing Perpetuity:
C
PV 
rg
Growing Annuity:
C
PV 
rg
PV 
What about a (growing)
perpetuity
starting in year 4?
  1  g T 
1  
  if r  g
  1  r  
TC
if r  g
1 r
2 types of Time Value Formulae:
I: A single cash flow moved multiple time periods
II: Multiple cash flows moved a single time period
2-9
Quick Quiz
 In 1934, the first edition of a book described by
many as the “bible” of financial statement
analysis was published. Security Analysis by
Grham and Dodd has proven so popular among
financial analysts that it has never been out of
print.
According to an item in The Wall Street Journal, a
copy of the first edition was sold by a rare book
dealer in 1996 for $7,500. The original price of the
first edition was $3.37. What is the annually
compounded rate of increase in the value of the
book?
2-10
Future Values and Multiple Cash Flows
Example: Suppose your rich uncle offers to help pay for your
business school education by giving you $5,000 each year for the
next three years beginning today (year = 0). You plan to deposit this
money into an interest-bearing account so that you can attend
business school six years from today. Assume you earn 4.25% per
year on your account. How much will you have saved in six years?
Year
Cash Flow
Future Value
Factor
0
5,000
(1.0425)6 = 1.2837
$6418.39
1
5,000
(1.0425)5 = 1.2313
$6156.73
2
5,000
(1.0425)4 = 1.1811
$5905.74
Total Future Value
Future Value
$18,480.86
2-11
Present Value
 Want to be a millionaire? No problem!
Suppose you are currently 21 years old,
and can earn 10 percent on your money
(about what the typical common stock has
averaged over the last six decades - but
more on that later). How much must you
invest today in order to accumulate
$1 million by the time you reach age 65?
FV = PV × ( 1 + r )t  PV = FV / ( 1 + r )t
 FV = $1 million, r = 0.10, and t = 44 => PV = $15,092
2-12
Present Values: Multiple Periods
 Suppose you need $10,000 in three years. If you earn
5% each year, how much money do you have to invest
today to make sure that you have the $10,000 when
you need it?
 PV = $10,000 / (1.05)3 => PV = $8,638.38
 What is the maximum price you’d be willing to pay for
a promise to receive a $25,000 payment in 30 years?
You can invest your money somewhere else with
similar risk and make a 24% annual return.
 PV = $25,000 / (1.24)30 => PV = $39.38
2-13
Investing for More than One Period:
Present Values and Multiple Cash Flows
 Suppose your firm is trying to evaluate whether
to buy an asset. The asset pays off $2,000 at the
end of years 1 and 2, $4,000 at the end of year 3
and $5,000 at the end of year 4. Similar assets
earn 6% per year. How much should your firm
pay for this investment?
 Rule: Discount cash flows to the present, one set
of cash flows at a time and then add them up.
2-14
Year
Cash Flow
Present Value
Factor
1
2,000
1 / (1.06)
$1886.79
2
2,000
1 / (1.06)2
$ 1779.99
3
4,000
1 / (1.06)3
$ 3358.48
4
5,000
1 / (1.06)4
$ 3960.73
Total Present Value
Present Value
$10,985.73
2-15
Finding the Number of Periods
 Sometimes we will be interested in knowing how long it
will take our investment to earn some future value.
Given the relationship between present values and
futures value, we can also find the number of periods.
We can solve for the number of periods by rearranging
the following equation:
FV = PV × (1 + r)t  FV / PV = (1 + r)t
 ln(FV / PV) = ln (1 + r)t
 ln(FV) - ln(PV) = t × ln (1 + r)
 t = (ln(FV) - ln (PV)) / ln (1 + r)
2-16
Finding the Number of Periods
 How long would it take to double your
money at 5%?
t = (ln(FV) - ln (PV)) / ln (1 + r)
Approximately 14 years and 2 months
Rule of thumb: Rule of 72
 How long for your money to double at 9%?
 How long for your money to triple at 11%?
2-17
PV (Annuity) Calculation
 Assumes annuity payment occurs at the end of
the period.
 Cash flows of an annuity are all the same
 Period covered by the interest rate r must
correspond to the frequency of the annuity
payment
 The present value of an annuity of C dollars per
period for t periods when the rate interest rate is r
is :
C
PV(Annuity) =
r

1 
1 t
 (1 + r ) 
2-18
Present Value of an Annuity: Example
 Suppose you need $25,000 each year for
business school. You need the first $25,000 at
the end of 12 months and the second $25,000 at
the end of 24 months. If you can earn 8% per year
on your money how much do you need today to
be able to afford business school?
2-19
Future Value of an Annuity


C
FV(Annuity) = (1 + r )t - 1
r
Suppose you plan to retire ten years from today.
You plan to invest $2,000 a year at the end of each
of the next ten years. You can earn 8% per year on
your money. How much will your investment be
worth at the end of the second year? How much
will it be after ten years?
2-20
Example: Finding t
 Q.
Suppose you owe $2000 on a VISA card, and the
interest rate is 2% per month. If you make the
minimum monthly payments of $50, how long will it
take you to pay it off?
 A.
A long time:
$2000 = ($50/0.02) x [1 - ( 1 / 1.02)t ]
$2000 = 2500 × [1 - (1 / 1.02)t ]
$2000/$2500 -1 = - (1 / 1.02t) => - 0.2 = - (1 / 1.02t)
0.2 × 1.02t = 1 => 1.02t = 5 => t = ln(5) / ln(1.02)
81 months, or about_______
6.5 years
t = ________
2-21
Perpetuities
 A perpetuity is an annuity in which the stream of
cash flows continues forever.
 Suppose we are examining a perpetuity that costs
1,000 and offers a 12% rate of return. The cash
flow each year is $1,000*0.12 = $120. More
generally, the present value of a perpetuity
multiplied by the rate of interest must equal the
cash flow:
C
PV(Perpetuity) =
r
2-22
 The present value of a perpetual cash flow stream has a
finite value (as long as the discount rate, r, is greater than 0).
Here’s a question:
How can an infinite number of cash payments have a finite
value?
 Here’s an example related to the question above. Suppose
you are considering the purchase of a perpetual bond. The
issuer of the bond promises to pay the holder $100 per year
forever. If your opportunity rate is 10%, what is the most you
would pay for the bond today?
 One more question: Assume you are offered a bond identical
to the one described above (no principal repayment, just
interest payments), but with a life of 50 years. What is the
difference in value between the 50-year bond and the
perpetual bond?
2-23
Preferred Stock as a Perpetuity
 Preferred stock is an example of a perpetuity.

The holder of preferred stock is promised a fixed cash dividend every
period (usually quarter). It is called preferred because the dividend is
paid before common stock dividends but after interest payments.
 Suppose GM wants to sell preferred stock at $100
per share. A very similar issue of preferred stock
outstanding has a price of $40 per share and
offers a dividend of $1 every quarter.
 What dividend will GM have to offer if the
preferred stock is to sell for $100?
 P2=C2/r  $40=1/r  r=0.025  P1=C1/r  $100=C1/0.025
 C1 = $2.50
2-24
Relation between annuities and perpetuities
How to remember formulae for
annuities?
Difference between 2 perpetuities!!!
PV(annuity) = C/r minus discounted C/r
FV(annuity) = future value C/r minus C/r
Draw timelines!
2-25
Growing Annuities and Perpetuities
 Cash flows grow g % per time period
 C = cash flow in first time period (t = 1)
 If r = g then PV = TC / 1+r
 Example: What is the PV of a $10 payment,
growing at 3% per year, for 4 years, with r = 10%?
 For a perpetual stream, growing at 3%, we
get: C / (r - g) = 10 / (0.07) = $142.86
2-26
Comparing Interest Rates:
The Effect of Compounding
 Stated or quoted rate: The annual rate before
considering any compounding effects, such as
10% compounded semiannually.
 Effective Annual Rate (EAR): The rate, on an
annual basis, that reflects compounding
effects, such as 10% compounded semiannually gives an effective rate of 10.25%.
2-27
Effective Annual Rates
 Why is it important to work with EARs? Suppose
you are interested in buying a new car. You have
shopped around for loan rates and come up with
the following three rates:
• Bank A:
• Bank B:
• Bank C:
12% compounded monthly
12% compounded quarterly
12.25% compounded annually
 Which is the best rate? We use effective annual
rates to compare the above lending rates.
2-28
Calculating EARs
m
 quoted rate 
EAR = 1 +
-1

m


What is the EAR for 12% compounded quarterly?
Step 1: Divided the quoted rate by the number of
times that interest is compounded during the year.
Step 2: Add 1 to the result and raise it to the power of
the number of times interest is compounded during
the year.
Step 3: Subtract 1 from your answer in Step 2.
2-29
Computing Present Values Using
EARs
 What is the present value of $100 to be
received at the end of two years at 10%
compounded quarterly?
 Step 1: Calculate the effective annual rate:
EAR=(1+(0.10/4))4 - 1=10.38%
 Step 2: Calculate the present value of the cash
flows.
PV = 100 / (1.1038)2 = $82.07
2-30
Annual Percentage Rates (APRs)
 Annual Percentage Rate: The rate per period
times the # of periods per year, making it a
quoted or stated rate.
 What is the annual percentage rate if the interest
rate is 1.25% per month?
 Example: If you look at the credit agreement for
your credit card, you will see that an annual
percentage rate is charged. But what is the
actual rate you pay on such a card if you do not
make your payment?
2-31
APRs and EARs
 An APR of 18% with monthly payments is 0.015 or
1.5% per month. What is the EAR?
EAR = (1 + (0.18/12))12 - 1 = 19.56%
2-32
Compounding Periods, EARs, and
APRs
Compounding
period
 Year
 Quarter
 Month
 Week
 Day
 Hour
 Minute
Number of times
compounded
1
4
12
52
365
8,760
525,600
Effective
annual rate
10.00000%
10.38129
10.47131
10.50648
10.51558
10.51703
10.51709
2-33
Converting Interest Rates: Summary
Rule: Convert Interest Rate to match
the Cash Flow Periods
Compounding: APR or EAR?
Periodic rate (i.e., monthly): APR / m
m = number of periods per year
APR = periodic rate × m
Effective Annual Rate:
EAR = [1 + (APR/m)]m – 1
EAR = [1 + (periodic rate)]m – 1
2-34
Loan Payments
 You have decided to buy a new four-wheel drive sports
vehicle and finance the purchase with a 10-year loan.
The loan is for $33,500. Interest starts accruing when
the loan is taken. The first loan payment is one-month
after the interest starts accruing. The interest rate on
the loan is 8.5% (APR) per year for the ten-year period.





What type of security is the series of loan payments?
What is the present value of the loan?
What discount rate should be used in the present value calculation?
Calculate the monthly loan payment.
How much have you paid off after 2 months?
Example: Cheap Financing or Rebate?
SALE! SALE!
5%* FINANCING OR $500 REBATE
FULLY LOADED MUSTANG
only $10,999
Option 1: Rebate:
 
C
$10,499 
 1
0.00833
C  $338.77
1
(1 0.00833) 3 6

Option 2: 5-% Financing:
*5% APR on 36 month loan.
If PNC Bank is offering 10% car
loans, should you choose the
5% financing or $500 rebate?
2-35
 
C
 1
0.00417
C  $329.65
$10,999 
1
(1 0.00417) 3 6
CONCLUSION: USE FINANCING DEAL
